Maxwell s equations. in differential form. r J

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1 Bacgoud Maell s equatios C 46/566 Guided Wave Optics Mawell s equatios i diffeetial fom B t D H t B D ρ J Faaday s law Ampee s law No mag. moopoles Gauss law lectic field [V/m] H Magetic field [A/m] D lectic flu desity [C/m ] B Magetic flu desity [Wb/m ] J lectic cuet desity [A/m ] ρ lectic chage desity [C/m 3 ] Cul [/m] Divegece [/m] If you ca t tasfom these to itegal fom i 3 secods, go eview. Robet R. McLeod, Uivesity of Coloado Polloc ad Lipso, Chapte 7

2 Bacgoud Maell s equatios C 46/566 Guided Wave Optics Costitutive elatios Iteactio with matte D ε t ε ( t τ ) ( τ ) dτ ε ( t) ε δ ( t) ε ε ( τ ) ε ε I ε ε( τ ) Dispesive & aisotopic Aisotopic Isotopic t Nomagetic B µ µ ( t τ ) H( τ ) dτ µ H J σ Ohm s Law ε Pemittivity of fee space [F/m] ε Dielectic costat, elative pemittivity µ Pemeability of fee space 4 π -7 [H/m] µ Relative pemeability σ Coductivity [Ω/m] Robet R. McLeod, Uivesity of Coloado 8

3 Bacgoud Maell s equatios C 46/566 Guided Wave Optics Moochomatic fields pad all vaiables i tempoal eigefuctio basis f j ω ω t ( t) f ( ) e dω Fouie Tasfom π j t Re( e ω ) Moochomatic fields tasfom lie timedomai fields fo liea opeatos d dt jω Removes all time-deivates. H B D ρ jω B jω D J Moochomatic Mawell s equatios. Robet R. McLeod, Uivesity of Coloado 9

4 Bacgoud Maell s equatios C 46/566 Guided Wave Optics Moochomatic costitutive elatios The easo fo usig the moochomatic assumptio t D ε ε ( t τ ) ( τ ) dτ D ε ε ( ) Covolutio t µ ( t τ ) H Multiplicatio ω B µ ( τ ) dτ B µ µ ( ) H ω ε jωt ( ω) ε ( t) e dt jωt µ ( ω) µ ( t) e dt Ivese Fouie Tasfom. Note that ε is ow f(ω) & ot f(t). If ε is ot costat i ω, it causes dispesio of pulses. ε µ Coditios fo lossless mateials deived fom Poytig vecto (et) ε µ ε is the Hemitia cojugate: ε * ji ε ij Robet R. McLeod, Uivesity of Coloado

5 Bacgoud Maell s equatios C 46/566 Guided Wave Optics Comple dielectic teso Fo coductive mateials H jω D J jωε ε σ j jωε ( ε σ ) ωε Ampee s law Costitutive elatios Goup tems Comple dielectic teso Fom this poit o the dielectic teso will be tae to be comple via this defiitio. Robet R. McLeod, Uivesity of Coloado

6 Bacgoud Maell s equatios C 46/566 Guided Wave Optics Poytig vecto Powe flow P H Istataeous powe flow P T T P Re Re ( P) dt ( H ) Time-aveaged powe flow Algeba left out hee Defie comple vecto P H Real pat of P is < P > P Powe pe uit aea [W/m ] Robet R. McLeod, Uivesity of Coloado

7 Modal aalysis of plaa slab waveguides M bouday coditios C 46/566 Guided Wave Optics Bouday coditios (/) Fields at shap chage of mateial These ae deived fom Mawell s equatios via applicatio of divegece ad cul elatios at a shap dielectic bouday. Loo them up i you favoite &M boo. I the absece of suface chage o cuet H D B,,,, H D B,,,, Cosevatio of tasvese electic ad magetic fields Cosevatio of omal electic ad magetic flu desities Medium Medium Uit vecto omal to bouday Uit vecto paallel (o tagetial) to bouday Robet R. McLeod, Uivesity of Coloado 3

8 Modal aalysis of plaa slab waveguides M bouday coditios C 46/566 Guided Wave Optics Bouday coditios (/) Deivatives of &H at shap chage of mateial H, These imply that the tagetial, deivatives of the tagetial,, field compoets must be, H, cotiuous at the bouday. H, H, Buildig similaly fom the ow bouday coditios o fields usig Mawell s equatios, we ca fid the followig: C D H C H C C? D D C D C C whee C implies cotiuity, D implies discotiuity, ad? implies that eithe ae possible. The impotat poit is that H is i geeal moe cotiuous acoss puely dielectic boudaies, motivatig the use of the magetic wave equatio to solve cetai poblems. Robet R. McLeod, Uivesity of Coloado Mäz, Sectio.. 4

9 Bacgoud Waves i space C 46/566 Guided Wave Optics Vecto wave equatio limiate all fields but jω B jωµ H ω µ D ω ε µ ε Tae cul of Faaday s law Magetic costitutive Ampee s law lectic costitutive ε ( ) ˆ y yˆ Moochomatic vecto W Wave umbe of fee space ω/c π/λ [/m] c Speed of light i vacuum µ ε [m/s] z zˆ Vecto idetity Vecto vs. scala Laplacia Robet R. McLeod, Uivesity of Coloado 5

10 Bacgoud Waves i space C 46/566 Guided Wave Optics Scala wave equatio Valid i slowly-vayig dielectics D ε ( ) Substitute i idetity If o fee chage ( ε ) ε ( ε ) ε ( ε ) ( lε ) Costitutive elatio (isotopic) Chai ule Solve fo div Sometimes witte this way ε ε ε Isotopic, vecto W. RHS is ofte quite small, leadig to... Scala simplificatio Wave umbe i dielectic ε [/m] Robet R. McLeod, Uivesity of Coloado 6

11 Bacgoud Waves i space C 46/566 Guided Wave Optics Solutio of scala W I Catesia space Scala W pad vecto Laplacia i Catesia coodiates ˆ yˆ zˆ y z y y z z [ ˆ yˆ zˆ ] Vecto equatio ow sepaates ito 3 scala equatios Robet R. McLeod, Uivesity of Coloado 7 y Note Laplacias ae ow scala Laplacia Solutios ae popagatig waves: j A e A ˆ is comple amplitude (phaso) of plae wave is the wave-vecto, a etemely impotat quatity. Diectio of is diectio of popagatio. It is paallel to Poytig vecto i isotopic space. It s magitude is π / λ π f whee f is spatial fequecy i uits of [/m] aalogous to tempoal fequecy i uits of [/s]. Note that oe solutio ( ) is disallowed by Gauss Law. z

12 Bacgoud Waves i space C 46/566 Guided Wave Optics λ λ λ si θ Spatial fequecy Basis of Fouie optics ic ic λ θ ic ẑ θ tas tas vacuum ˆ f si θ θ ˆ ic si tas λ λ λ π Spatial fequecy i [/m] π f π λ π siθ λ ic π siθ λ tas Wave umbe i [/m] Multiply by h ad ow the vectos epeset quatummechaical mometum. The icidet paticle sties a bouday omal to z which chages its mometum i z but ot i,y. Theefoe, tasvese mometum is coseved. Robet R. McLeod, Uivesity of Coloado 8

13 Bacgoud Waves i space C 46/566 Guided Wave Optics Sell s Law fom cosevatio of peiodicity o the bouday ẑ tas z θ tas tas ˆ ic θ ic ic ic ˆ tas ˆ e ˆ ˆ z zˆ j( ωt ic tas ) Siθ Bea both wave vectos ito omal ad tasvese compoets. ic Siθ tas citatio o bouday. Tasvese wave vecto coseved. ic ic ic Siθ Siθ Siθ ic ic ic tas tas Siθ tas Siθ tas Siθ tas tas Sell s Law Robet R. McLeod, Uivesity of Coloado 9

14 Fouie popagatio amples C 46/566 Guided Wave Optics Isotopic efactio a..a. Sell s law gives ay diectios at bouday Real space z ẑ θ tas tas ic θ ic ˆ Fouie space z Glass Ai.5.5 θ ic tas ic t z -.5 θ efl efl Real pat Imag. pat Robet R. McLeod, Uivesity of Coloado ic tas efl Siθ Siθ ic Siθ tas efl

15 Fouie popagatio amples C 46/566 Guided Wave Optics Total iteal eflectio a..a. evaescet waves Real space ẑ α z tas e ( j ˆ α zˆ) z Ai Glass ic θ ic θ efl efl ˆ Mometum space z.5 Glass Ai.5 θ ic ic α t θ efl efl Real pat Imag. pat Robet R. McLeod, Uivesity of Coloado ic efl Siθ Siθ ic efl

16 Ray aalysis of waveguides Fesel eflectios C 46/566 Guided Wave Optics Fesel Coefficiets Amplitude ad phase of waves at bouday NOT THAT ANGL θ IS FROM NORMAL. W WILL OFTN US θ MASURD FROM TO BOUNDARY! s ( seecht ) / T / T i Si Si ( θ θ ) ( θ θ ) t T t i Siθ Si ( θ θ ) p ( paallel ) / TM / TM i Ta Ta ( θ θ ) ( θ θ ) Ca t TM t i Si Siθ ( θ θ ) Cos( θ θ ) Robet R. McLeod, Uivesity of Coloado Saleh & Teich 6.

17 Ray aalysis of waveguides Fesel eflectios C 46/566 Guided Wave Optics Fesel Coefficiets Special values t TM TM t T < T θ B R Nomal icidece θ B Ta Bewste s agle No eflectio fo TM. π θ θ i θ θ t t Physical itepetatio of θ B : Dipoles ecited i ca ot adiate i diectio of eflected wave whe it is to diectio of popagatio i egio Robet R. McLeod, Uivesity of Coloado 3

18 Ray aalysis of waveguides Fesel eflectios C 46/566 Guided Wave Optics Fesel coefficiets Goos-Häche phase shift Fo >, TIR is possible. Beyod the citical agle the eflectio has uity magitude with a shift of phase. T cosθ cosθ cosθ cosθ cosθ cosθ Fesel fomula si si θ θ Sell s Law ep cosθ j cosθ [ jφ ] T j si si θ θ θ > θ C Si Num. ad de. same mag, opp phase taφ T si θ cosθ Goos-Häche phase fo TM taφ TM si θ cosθ By same deivatio fo TM Diffes oly i ε /ε This is the taditioal fom fo these epessios. Note, howeve: Phase shift o eflectio is φ Robet R. McLeod, Uivesity of Coloado 4

19 Ray aalysis of waveguides Fesel eflectios C 46/566 Guided Wave Optics Fesel Coefficiets Phase ad TIR > t TM TIR t T T θ C Si TM π θ B θ C φ TM φ T Goos-Häche phase-shift φ π fo fo > < Phase of T electic field o eflectio. TM has the opposite (same) sig < (>) θ B. Robet R. McLeod, Uivesity of Coloado 5

20 Bacgoud Ray ad eioal equatios C 46/566 Guided Wave Optics The ay equatio Appo. solutio of M whe () is slow js ( ) ( ) ( ) e S ( ) y z ( ) ε ( ) ( ) Assume slowly vayig amplitude ad phase S S y z.g. plae wave ( ) y S z.g. spheical wave Substitute ito isotopic scala wave equatio ad etai lowest tems Ray equatio eductio of Mawell s equatios. Coutous of S() at multiples of π Ray cuve to S() () S ( )ds S() Optical path legth [m] Robet R. McLeod, Uivesity of Coloado 6 B A

21 Bacgoud Ray ad eioal equatios C 46/566 Guided Wave Optics The eioal equatio A equatio fo evolutio of ay tajectoy Tae squae-oot of ay equatio, d d S( ) ( ) sˆ ( ) ds ds the tae deivative i s, sds ˆ d d d [ S( )] ( ) ds ds ds d d ds ad chai ule fo, d ds [ S( )] [ S( )] fially apply aothe idetity. ( ) [ S( ) S( )] d ds S ( ) ( ) [ S( )] s Paametic distace alog ay [m] ( s) Ray tajectoy [m] Ide of efactio, phase velocity c/ Robet R. McLeod, Uivesity of Coloado Saleh & Teich.3 7 d ds ( ) ( ) ( ) Ray [ ( )] ( )